Stochastic approach and irreversible chemical reactions in small systems

In a previous work it has been established that a quantum Smoluchovski equation can be used to describe the irreversible behavior of small systems. Here we use this equation to study the motion of a particle injected in an existing symmetric double well potential. The dynamics is investigated via the chemical rate. Explicit relations are given for the equilibrium value of this quantity and for the relaxation time describing the evolution towards the equilibrium. The coupling between the system and a thermostat is discussed. The irreversibility is described by a monotonic increasing function of time, which tends at the equilibrium towards a quantity that we may consider as the thermodynamic entropy. We compare our approach with the Kramers result and with those obtained in the system + reservoir approaches. We do not compare our approach to a precise experimental system but we check that the predicted results have the expected order of magnitude in the case of a proton moving in a existing double well potential. A comparison with the standard description based on the Schrِdinger equation is presented; some ingredients are the same but the time evolution is obviously not the same. Although the coupling with a bath of oscillators is not in the main scope of this paper we briefly show how this bath may introduce an additional friction constant and a functional that we may interpret as giving the reorganization energy of the medium.